Armen G. Khachaturyan

Three-dimensional field model and computer modeling of martensitic transformations

A three-dimensional (3D) continuum stochastic field kinetic model of martensitic transformations which explicitly takes into account the transformation-induced elastic strain is developed. The model is able to predict the major structural characteristics of martensite during the entire transformation including nucleation, growth and eventually formation of internally twinned plates which are in thermoelastic equilibrium with the parent phase. No a priori constraints are made on the possible configurations and sequences of structural patterns formed by orientation variants of the martensite. 3D computer simulations are performed for a generic cubic → tetragonal martensitic transformation in a prototype crystal which is elastically isotropic and elastically homogeneous. The simulations predict that (i) nucleation of martensite in a perfect crystal occurs collectively to accommodate the coherency strain, e.g. the critical nuclei are formed by two internally twinned orientation variants; (ii) the ultimate structure consists of plate-like martensite and retaining parent phase. The martensitic plates consist of twin-related platelets of two orientation variants and the habits of the plates meet the invariant plane requirement. These simulation results are in good agreement with experimental observations.

Nanoscale phase field microelasticity theory of dislocations: model and 3D simulations

The first Phase Field model of evolution of a multi-dislocation system in elastically anisotropic crystal under applied stress is formulated. The model is a modification and extension of our Phase Field Microelasticity approach to the theory of coherent phase transformations. The long-range strain-induced interaction of individual dislocations is calculated exactly and is explicitly incorporated in the Phase Field formalism. It also automatically takes into account the effects of “short-range interactions”, such as multiplication and annihilation of dislocations and a formation of various metastable microstructures involving dislocations and defects. The proposed 3-dimensional Phase Field model of dislocations does not impose a priori constraints on possible dislocation structures or their evolution paths. Examples of simulation of the FCC 3D system under applied stress are considered.

Three-dimensional phase field model of proper martensitic transformation

Abstract The Phase Field Microelasticity theory is developed for proper multivariant martensitic transformations. The model is based on the exact solution of the elasticity problem in the homogeneous modulus approximation. The model takes into account the transformation-induced coherency strain and provides for the strain compatibility throughout the system. Computer simulations are performed for a dilatationless cubic→tetragonal martensitic transformation and for the transformation with parameters corresponding to a martensitic transformation Fe–31%Ni alloy. The development of the martensitic transformation through nucleation, growth and coarsening of orientation variants is simulated at different levels of undercooling. The simulated martensitic structure has a complex polytwinned morphology. Simulation demonstrates that the presence of a non-zero volumetric component in the transformation strain in the Fe–31%Ni system significantly affects the martensitic transformation.

Field kinetic model and computer simulation of precipitation of L12 ordered intermetallics from f.c.c. solid solution

A phenomenological stochastic field kinetic model of coherent precipitation of L12 ordered intermetallics from a disordered f.c.c. solid solution has been developed. It explicitly takes into account both the lattice misfit strain and the four types of antiphase domains formed as a result of the L12 ordering. Based on the concentration wave representation of the ordered state, a coarse grain approximation of the non-equilibrium free energy functional of concentration and long-range order parameters has been formulated. Its relation to the microscopic free energy model has been discussed. Precipitation kinetics and microstructural development of γ′ phase in Ni–Al have been investigated by two-dimensional computer simulations based on this model. The results reveal new features which are attributed to the L12 ordering. The ordered nature of the precipitates changes the coarsening mechanisms and kinetics and greatly affect the morphology of the mesoscopic microstructure.

Kinetics of strain-induced morphological transformation in cubic alloys with a miscibility gap

Morphological evolutions controlled by a transformation-induced elastic strain during a solid state precipitation are systematically investigated using a prototype binary alloy as a model system. A computer simulation technique based on a microscopic kinetic model including the elastic strain effect is developed. Without any a priori assumptions concerning shapes, concentration profiles and mutual positions of new phase particles, various types of coherent two-phase morphologies such as basket-weave structures, sandwich-like multi-domain structures, precipitate macrolattices and GP zones are predicted. A wide variety of interesting strain-induced kinetic phenomena are observed during development of the above microstructures, including selective and anisotropic growth, reverse coarsening, particle translational motion, particle shape transition and splitting. In spite of all simplifications of the model, most of the simulation results are confirmed by experimental observations in various alloy systems, indicating that this kinetic model can be efficiently used for understanding, interpreting and predicting structural evolutions in real alloys.

Adaptive ferroelectric states in systems with low domain wall energy: Tetragonal microdomains

Ferroelectric and ferroelastic phases with very low domain wall energies have been shown to form miniaturized microdomain structures. A theory of an adaptive ferroelectric phase has been developed to predict the microdomain-averaged crystal lattice parameters of this structurally inhomogeneous state. The theory is an extension of conventional martensite theory, applied to ferroelectric systems with very low domain wall energies. The case of ferroelectric microdomains of tetragonal symmetry is considered. It is shown for such a case that a nanoscale coherent mixture of microdomains can be interpreted as an adaptive ferroelectric phase, whose microdomain-averaged crystal lattice is monoclinic. The crystal lattice parameters of this monoclinic phase are self-adjusting parameters, which minimize the transformation stress. Self-adjustment is achieved by application of the invariant plane strain to the parent cubic lattice, and the value of the self-adjusted parameters is a linear superposition of the lattice c...

Three-dimensional phase field model of low-symmetry martensitic transformation in polycrystal: simulation of ζ′2 martensite in AuCd alloys

A three-dimensional phase field model of the martensitic transformation that produces a low symmetry phase in polycrystals is developed. The transformation-induced strain mostly responsible for the specific features of the martensitic transformation is explicitly taken into account. The high computational efficiency of the model turns out to be almost independent of the complexity of the polycrystal geometry. An example of the cubic→trigonal transformation in AuCd alloys producing ζ′2 martensite is considered. The development of the transformation through nucleation, growth and coarsening of orientation variants is simulated for both single crystal and polycrystalline materials. The effect of an external load on the martensitic microstructure in the polycrystalline material is studied. It is shown that the elastic coupling between different transformed grains of the polycrystal drastically affects the microstructure and its response to the applied stress. The obtained self-accommodating morphologies of the multivariant martensitic structure are in agreement with those observed in the experiments.

THREE-DIMENSIONAL PHASE FIELD MODEL AND SIMULATION OF MARTENSITIC TRANSFORMATION IN MULTILAYER SYSTEMS UNDER APPLIED STRESSES

Abstract The phase field microelasticity theory is used to formulate a three-dimensional phase field model of a multivariant martensitic transformation under external load. The model is based on the exact solution of the elasticity problem in the homogeneous modulus approximation. The transformation-induced coherency strain and applied stress are explicitly taken into account. Computer simulations are performed for a generic cubic→tetragonal martensitic transformation in a multilayer system consisting of alternating active and inert layers. The development of the martensitic transformation through nucleation, growth and coarsening of orientation variants is simulated at different levels of the applied stress. The simulated martensitic structure has a complex polytwinned morphology. The simulation predicted such effects as the formation of texture and the stress-induced transformation that are in a general agreement with the experimental observations. The simulation produced realistic stress–strain hysteresis loops, which, in principle, can be used for the formulation of the constitutive equations of the macroscopic mechanics for the active system.

Ferroelectric solid solutions with morphotropic boundaries: Vanishing polarization anisotropy, adaptive, polar glass, and two-phase states

The generic case of a ferroelectric solid solution is considered wherein different symmetry phases located at opposing ends of the diffusionless phase diagram are separated by a morphotropic boundary (MB). It is shown that the Landau theory of weak first-order phase transformations automatically predicts vanishing of the anisotropy of polarization, continuity of thermodynamic properties, and a drastic decrease in domain wall energy near the MB line that results in the formation of adaptive ferroelectric nanodomain states. Low-resolution diffraction from these adaptive states acquired at the coherence lengths of elastic x-ray or neutron scattering probes will produce the same diffraction pattern as attributed to monoclinic (MA,MB,MC) phases. It is further shown that the electric- or stress-field-induced reconfiguration of these adaptive nanodomain states results in a softening of the piezoelectric, elastic, and dielectric properties near the MB line. In addition, the spherical degeneration of the polarization direction, reflecting the decoupling of the polarization from the crystal lattice at the MB, also predicts the formation of a polar glass state whose properties should be similar to the special properties of amorphous ferromagnets. In particular, the vanishing of the polarization anisotropy at the MB should result in ferroelectric domains with irregular shapes exhibiting high configurational sensitivity to external forces. The theory further predicts that two tricritical points will occur on the line of paraelectric→ferroelectric transitions and it is shown that all two-phase equilibrium lines of the diffusionless phase diagram—including the MB line—must be replaced by two-phase fields. Within these two-phase fields, the adjacent ferroelectric-ferroelectric and paraelectric-ferroelectric phases coexist. Possible topologies of the equilibrium MB phase diagram illustrating these two-phase equilibrium fields are computed and discussed.The generic case of a ferroelectric solid solution is considered wherein different symmetry phases located at opposing ends of the diffusionless phase diagram are separated by a morphotropic boundary (MB). It is shown that the Landau theory of weak first-order phase transformations automatically predicts vanishing of the anisotropy of polarization, continuity of thermodynamic properties, and a drastic decrease in domain wall energy near the MB line that results in the formation of adaptive ferroelectric nanodomain states. Low-resolution diffraction from these adaptive states acquired at the coherence lengths of elastic x-ray or neutron scattering probes will produce the same diffraction pattern as attributed to monoclinic (MA,MB,MC) phases. It is further shown that the electric- or stress-field-induced reconfiguration of these adaptive nanodomain states results in a softening of the piezoelectric, elastic, and dielectric properties near the MB line. In addition, the spherical degeneration of the polarizat...

COMPUTER SIMULATION OF STRUCTURAL TRANSFORMATIONS DURING PRECIPITATION OF AN ORDERED INTERMETALLIC PHASE

Abstract A computer the simulation of kinetics of precipitationof an ordered intermetallic phase from a disordered solid solution in a model binar two-dimensional alloy is considered. All diffusion processes, the concentrational delamination (clustering), ordering, antiphase domain boundary movement and coarsening, are described by the microscopic kinetic equations. The ordering transition of the first and second kinds are considered. It is shown that the conventional precipitation mechanism through nucleation and growth of an equilibrium ordered phase occurs only in a very narrow “strip” of the two-phase field in the phase diagram. In the remaining part, the decomposition always starts from a congruent ordering, which produces a transient nonstoichiometric ordered single-phase state with the same composition as the parent disordered phase and the same symmetry as the product intermetallic phase. Decomposition of the transient ordered phase occurs predominantly at the antiphase domain boundaries (APBs), which results in a two-phase morphology with layers of disordered films separating antiphase domains of the ordered phase. This decomposition is a result of a concentration instability on APBs, which is more substantial than the conventional homogeneous spinodal instability. Further morphological evolution after decomposition is controlled by coarsening, which reduces the order/disorder interfacial area. The predicted precipitation mechanism through a formation of a single-phase transient ordered state is general. It is expected in the most part of a two-phase field of a phase diagram of any alloy systems with intermetallic precipitates related to the parent one by ordering.